The comments by Andreas and Anton give you the answer already to your specific question. Let me give a more general answer, since your question is very representative of a whole class of examples.

The condition that $x_n = 1 \Rightarrow x_{n+1} = 0$ is a *Markov* condition: the value of $x_{n+1}$ is restricted by the value of $x_n$. In your case you are considering all sequences in $\{0,1\}^\mathbb{N}$ such that the symbol $1$ cannot follow itself; one could also consider more symbols and more complicated restrictions, such as "every occurrence of $2$ can only be followed by $0$ or $2$, but not $1$". See http://en.wikipedia.org/wiki/Subshift_of_finite_type for more details.

Subshifts of finite type (abbreviated SFTs) are also called topological Markov chains, and can be presented in terms of a transition matrix, as described in that Wikipedia article. The logarithm of the largest eigenvalue of the transition matrix is an important quantity called the *topological entropy* of the SFT.

When you construct a subset of the Cantor set as in your question, the topological entropy turns out to be directly related to the Hausdorff dimension: namely Hausdorff dimension is topological entropy divided by $\log \lambda$, where $\lambda$ is the contraction ratio at each step of the construction of the Cantor set.

We wrote a more detailed description of this in Pesin & Climenhaga, "Lectures on fractal geometry and dynamical systems", or you can find many parts of it in most standard textbooks on dynamical systems.